Let
... then we say that
A matrix that doesn't have an inverse is called singular or non-invertible.
The Gauss-Jordan elimination method can be used to find the inverse of a matrix. Here's a step-by-step process:
Augment the Matrix with the Identity Matrix.
Apply Row Operations to Reach RREF: Use the Gauss-Jordan elimination method to transform the given matrix into the reduced row echelon form (RREF). The goal is to transform the left side of the augmented matrix into the identity matrix.
Use elementary row operations, including swapping rows, multiplying a row by a nonzero scalar, and adding/subtracting a multiple of one row from another row.
Note that this method only works for square matrices that are invertible.
If the given matrix is not invertible (i.e., it has no inverse), the process will reveal this fact by leading to a row of zeros or inconsistencies in the augmented matrix.
Suppose we want to find the inverse of the following matrix:
Augment with the Identity Matrix:
Row Operations:
Extract the Inverse:
The inverse of
Existence of the Inverse:
The inverse of a matrix
A non-zero determinant indicates that the matrix represents a transformation that doesn't "collapse" space; hence, it can be reversed.
Transposition of an Inverse:
If
Transposing a matrix and finding its inverse are reversible operations, so the order in which you perform them doesn't matter.
Inverse of a Scalar Multiple:
If
Scaling a matrix by a factor
Let A and B be invertible matrices, then AB is also invertible and
Intuitively, you can solve the an equation of the form
You can use the same method for solving system of linear equations of the form...
Where A is an invertible matrix.